A new perspective on Functional Integration

TL;DR

This paper introduces a general theorem for functional integration on manifolds, leading to a new method for solving Schrödinger-type equations directly through path integrals with rigorous proofs and multiple applications.

Contribution

It develops a broad functional integration framework on manifolds that generalizes existing methods and applies to non-time-invariant systems, with rigorous proofs and multiple applications.

Findings

Derived a general functional integral formula with applications.

Computed semiclassical expansions for the functional integrals.

Provided seven non-trivial applications demonstrating the method's effectiveness.

Abstract

The core of this article is a general theorem with a large number of specializations. Given a manifold $N$ and a finite number of one-parameter groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots, X_{(d)} $, we obtain, via functional integration over spaces of pointed paths on $N$ (paths with one fixed point), a one-parameter group of functional operators acting on tensor or spinor fields on $N$. The generator of this group is a quadratic form in the Lie derivatives $\La_{X_{(\a)}}$ in the $X_{(\a)}$-direction plus a term linear in $\La_Y$. The basic functional integral is over $L^{2,1}$ paths $x: {\bf T} \ra N$ (continuous paths with square integrable first derivative). Although the integrator is invariant under time translation, the integral is powerful enough to be used for systems which are not time translation invariant. We give seven non trivial applications of the basic formula, and we compute its semiclassical expansion. The methods of proof are rigorous and combine Albeverio H\oegh-Krohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches we solve Schr\"odinger type equations directly, rather than solving first diffusion equations and then using analytic continuation.